Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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3
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some important proofs about adjoint operators [duplicate]

I was told that the formal adjoint of the gradient is the negative divergence. Let $A : H\to H$ be a bounded, linear operator, The adjoint of $A$, i.e. $A^*: H\to H$ satisfies \begin{equation*} ...
2
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0answers
32 views

Is it possible to define submanifold like this

Wikipedia offers the following definition for an (embedded) submanifold: An embedded submanifold (also called a regular submanifold), is an immersed submanifold for which the inclusion map is a ...
-3
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0answers
20 views

An compute of Riemannian geometry [on hold]

According to Einstein summation convention , $g_{ij}$ is metric tensor,and $f$ is a real function. Show that : $$ g_{jl}g^{ij}\frac{\partial f}{\partial x^i}g^{kl}\frac{\partial f}{\partial ...
8
votes
1answer
1k views

Vectors, Basis, Dual Vectors, Dual Basis and Tensors

I'm trying to understand tensors and I know they have something to do with the basis and the dual basis of a vector space and a dual space. First I will give a concrete example to make clear what I ...
2
votes
1answer
48 views

Calculate the pushforward of smooth map between manifolds

Let $\Phi : GL(n)\rightarrow Sym(n)$ be defiened by $\Phi (A)=AA^T$. I can not see how to get the "right" pushforward, I.e I want help in understanding the pushforward $\Phi _*:M_I(n)\rightarrow ...
2
votes
1answer
16 views

A certain zeta function; or, the determinant of the Laplacian plus a constant on the circle

I am interested in a certain "zeta function," a meromorphic function of $s \in \mathbb{C}$ that depends on a real parameter $\alpha \neq 0$. It's defined for the real part of $s$ large by $$ ...
1
vote
1answer
48 views

Frechet mean of $k$ elements in the n-dimentional sphere.

The Frechet mean of $k$ elements $x_1, \ldots, x_k \in S = \{ x \in \mathbb{R}^n,\, \| x \| = 1 \}$ is defined as the $\text{arg}\underset{\|x\|=1}{\text{min}} \sum_{i=1}^n d^2(x_i,x)$. Where ...
4
votes
1answer
23 views

green's second identity application

I need to use the green's second identity in order to prove the following equality: $$ \int_{\mathbb{R}^2} \ln (\sqrt{x^2+y^2})\Delta f = -2\pi f(0)$$ where $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ ...
0
votes
1answer
79 views

A question about coadjoint orbit

If the coadjoint orbit $\Omega\subset \mathfrak{g^*}$ be contractible then prove that $\Omega$ is integral , i.e., $\int_C \omega\in \mathbb{Z}$ for every integral singular 2-cycle $C$ in $\Omega$, ...
0
votes
0answers
19 views

Proving the Leibniz Rule for Lie Derivatives of tensor fields.

I am learning some Differential Geometry on my own in preparation for a course I'm starting in October, and one of the exercises in the notes I'm using is to check that the Lie Derivative satisfies ...
0
votes
1answer
14 views

Metric compatibility of dual connection

Let $(M,g)$ be a Riemannian manifold with Levi Civita connection $\nabla$. Then $\nabla$ satisfies a compatibility condition: $(\nabla_ZX,Y)+(X,\nabla_ZY)=Z((X,Y))$ where $(\cdot,-)$ is a Hermitian ...
0
votes
2answers
48 views

Locus with line segments ratio constant.

$OAB$ is a rotating radial ray through origin $O$. Find a continuous curve through A and B so that quotient $OA/OB$ is constant, excluding Euclidean motion of rotation around $O$. A and B can also be ...
0
votes
0answers
15 views

Is this end-point map surjective

Consider the differential equation: $\frac{d U_s}{dt} = (a + w(s)b)U_s$ where $w$ is some unknown, smooth, real and bounded function on the interval $[0,T]$ and $a,b \in \mathfrak{su}(n)$. Let ...
0
votes
0answers
21 views

Preimage of a submanifold is a submanifold - Transversality

It is well known that if a smooth Map $f : M \to N$ between two smooth manifolds (finite dimensional) is transversal to a submanifold $L \subset N, L \pitchfork f$, than $f^{-1}(N)$ is a submanifold ...
1
vote
0answers
21 views

Symplectic matrix

I want to show that if $\lambda$ is a real eigenvalue of a symplectic matrix $A$ then its char poly is of the form $det(A-\mu id) = (\lambda-\mu)(\frac{1}{\lambda}-\mu) det(\hat{A}- \mu id) $ where ...
72
votes
9answers
3k views

Prove the theorem on analytic geometry in the picture.

I discovered this elegant theorem in my facebook feed. Does anyone have any idea how to prove? Formulations of this theorem can be found in the answers and the comments. You are welcome to join in ...
2
votes
3answers
117 views

uses of Riemannian geometry for questions not related to Riemannian geometry

Poincaré conjecture (whose formulation does not make use of notions from Riemannian geometry: "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere") was eventually proved using ...
0
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0answers
17 views

Show that the normals to a parameterized curve all pass through the z-axis

I've been asked to show that the normals to a parameterized surface given by: $x(u,v) = (f(u)cosv,f(u)sinv,g(u)), f(u) \neq 0, g'(u) \neq 0$ all pass through the z-axis. I've computed the normal to ...
1
vote
0answers
14 views

Jacobi determinant for high-dimensional sphere inversion

I need to find the Jacobi determinant for the unit sphere inversion in $\mathbb{R}^n$, i.e. the map given by $f(x) = \frac x {|x|^2}$ for $x\in \mathbb{R}^n$. The main problem is to figure out the ...
0
votes
1answer
30 views

Fitting an ellipse to a point with the first and second derivatives specified

I am trying to fit an ellipse to the end of a curve, $y(x)$, such that first and second derivatives of the curve, $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$, are preserved at the contact point, $(x,y)$, ...
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0answers
17 views

Show the first Chern class of a $U(1)$ bundle is integral.

I am working from John Baez's book: "Gauge Fields, Knots and Gravity". So I will stick to the notation used in that book. I am stuck at exercise 122 of part II (page 283), it reads: Show that if ...
0
votes
1answer
35 views

Stuck with a problem of calculus of variation in the proof that a minimizing curve is a geodesic

I'm reading the proof of the proposition that states that every minimizing curve is a geodesics when it is given an unit speed parametrization. In the proof appears the following quantity : $$ ...
0
votes
0answers
8 views

Bijective local isometry to global isometry

Suppose that I have a bijective local isometry $f: X \rightarrow Y$ where $X$ and $Y$ are length spaces. Can I show that $f$ is a global isometry? My thought is to consider a path $\gamma$ from $x$ to ...
0
votes
0answers
27 views

Smooth approximation to a continuous curve

Let $\gamma: [0,1] \rightarrow M$ be a continuous curve in a smooth manifold $M$. Is there a standard way to approximate $\gamma$ by a smooth curve? My thought was to look at every point $p$ where ...
4
votes
1answer
39 views

Zero Gaussian curvature and restriction estimates

Let $$ \left(\int_M|\hat{f}|^qd\mu(\xi)\right)^{1/q}\leq c||f||_{L^p(\mathbb{R}^n)} $$ be a restriction estimate for a hypersurface $M\subset\mathbb{R}^n,~1<p<\infty$ and $\mu$ a ...
0
votes
0answers
17 views

Image is not a manifold when considered as a subset: how is this possible?

Wikipedia offers two definitions of a submanifold: One is that it is the image of an immersion. But I can't make sense of the remark that " in general this image will not be a submanifold as a ...
1
vote
0answers
33 views

Inverse Function Theorem: is this true?

The inverse function theorem is usually stated as follows: Let $f:\mathbb R^n \to \mathbb R^n$ be a smooth map and let $x_0$ be a point such that $\det J_f (x_0) \neq 0$. Then there exists an open ...
1
vote
1answer
22 views

Is $\omega = \theta d\theta + zdz$ one 1-form in $S^2$ with cylindrical coordinates?

Take the $S^2$ sphere with cylindrical coordinates. We now that $\alpha = d\theta\wedge dz$ is the symplectic form of this manifold, with $\theta \in [0,2\pi)$ and $z \in (-1,1)$ . Following the same ...
0
votes
1answer
54 views

What are the prerequisites for Michael Spivak's monumental A Comprehensive Introduction to Differential Geometry?

What are the prerequisites for Michael Spivak's monumental A Comprehensive Introduction to Differential Geometry? In particular for volume 1? Are these 5 volumes self-consistent in the sense that a ...
1
vote
2answers
24 views

Show that the outward unit normal is smooth vector field

Exercise 2.1.8 of Guillemin and Pollack asks us to prove that if $X^m \subset \mathbb{R}^n$ is an embedded submanifold with boundary, then the outward unit normal to $\partial X$ is a smooth function ...
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vote
2answers
33 views

What rhumb lines of a torus are periodic $C^1$ curves?

This is a question coming from an old (French) geometry book. Take a 3D torus. Study the rhumb lines of the torus and find the ones that are periodic $C^1$ curves. In particular, it is mentioned that ...
0
votes
0answers
27 views

Extending bounded smooth curve in $\mathbb{R}^n$

If I have a smooth curve $\gamma:(0,1]\rightarrow\mathbb{R}^n$ such that the image of $\gamma$ is bounded can I extend this curve so that it smoothly approximate another curve whose endpoint agrees ...
3
votes
1answer
55 views

Is there a natural Riemannian structure on the total space of a vector bundle?

Suppose $B$ is a Riemannian manifold and $\pi: E \to B$ is a smooth vector bundle equipped with a metric. Is there a natural Riemannian metric on $E$, i.e. a bundle metric on $TE\to E$? It seems ...
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votes
0answers
15 views

Differential geometry qsn [on hold]

Find the curvature and torsion of the curves given by: $r = (a(3u-u^3),3au^2,a(3u+u^2))$ $r = a(1 + \cos u), a \sin u,2a \sin \frac u 2)$
1
vote
1answer
19 views

Laplace-Beltrami operator as sum of orthogonal projections

Let $M$ be a submanifold of $\mathbb R^l$ with the induced metric. Let $(\xi_\alpha)$ be the standard orthonormal basis on $\mathbb R^l$. For each $x \in M$, let $P_\alpha(x)$ the projection of ...
1
vote
1answer
37 views

Vector field left-invariant then also its respective flow?

I was wondering whether left invariance of a vector field $X$ to a respective Lie group $G$ (so $dL(a)(x)(X(x))= X(ax))$ is transfered to the respective flow defined by $\frac{d}{dt} \phi^{t}(x) = ...
2
votes
1answer
24 views

Evolution operator always in $SU(n)$?

Think about the evolution operator $U$ in Quantum Mechanics for finite dimensional systems. Then this operator satisfies an equation $$U'(t) = -iHU(t)..$$ Here, I assume that $H$ is ...
1
vote
0answers
16 views

Uniqueness of covariant derivative in Do Carmo

2.2 Proposition: Let $M$ be a differentiable manifold with an affine connection $\nabla$. There exists a unique correspondence which associates to a vector field $V$ along the differentiable curve ...
2
votes
2answers
73 views

Manifolds with a finite but not trivial fundamental group

I came across this nice result: Theorem: If $M$ is a connected smooth manifold with finite fundamental group, then its first de Rham cohomology is trivial: $$H^1_{dR}(M)=0.$$ However, I don't ...
1
vote
0answers
38 views

Identity regarding the components of a dual basis

This problem is from Robert Wald's "General Relativity." The problem is 4(b) from chapter 2. Let $Y_1\cdots Y_n$ be smooth vector fields on an $n$-dimensional manifold $M$ such that at each $p\in M$ ...
2
votes
0answers
32 views

Definition of a length metric.

Let $(X, d)$ be a metric space. Define the induced intrinsic metric $\widehat{d}$ as follows \begin{equation} \widehat{d}(x,y) = \inf_{\gamma} \sup_{t_0, \ldots, t_N} \sum_{i=1}^{N}d(\gamma(t_{i-1}, ...
3
votes
1answer
37 views

Pullback of a normal bundle

Consider $\Sigma$ a compact surface embedded into a compact 3-manifold, such that $\Sigma$ is diffeomorphic to $\mathbb{R}\mathbb{P}^2$ (real projective plane) and $\varphi:\mathbb{S}^2 \to \Sigma$ is ...
17
votes
1answer
1k views

What is magical about Cartan's magic formula?

Why is Cartan's magic formula $$\mathscr{L}_X\omega = i_Xd\omega + d(i_X\omega)$$ called "magic"? Should it be considered a highly surprising result? Does it "magically" prove several other ...
2
votes
1answer
337 views

Riemann tensor symmetries

The Riemann tensor has its component expression: ...
3
votes
1answer
19 views

limiting tangent line is parallel to asymptotic line

For a (infinitely, if necessary) differentiable curve $$ A(t) = (x(t), y(t)) $$ which diverges at $t_0 \in [-\infty,\infty] $, that is $$ \lim_{t \to t_0 } | x(t)^2 + y(t)^2 | =\infty $$ if there is ...
2
votes
0answers
23 views

Definition of $k$-precosymplectic manifold

A precosymplectic manifold of rank $2r$ is a triple $(M,\omega,\eta)$ where $M$ is a smooth manifold of dimension $2m+1$, $\omega$ is a closed 2-form on $M$ and $\eta$ is a closed 1-form on $M$ such ...
2
votes
0answers
36 views

Reference request: Tubular neighborhood theorem for non-closed manifolds via the exponential map.

Let $M\subset N$ be a submanifold. (Both $M$ and $N$ have no boundary, but $M\subset N$ need not to be closed as a subspace and none of them need to be compact) Choose a Riemannian metric on $N$ and ...
2
votes
0answers
24 views

Non-hyperbolic zeros of vector field

I'm wondering the following: Let $V$ be a vector field on a (compact Riemannian) smooth manifold $M$ with non-degenerate zeros. Let $p$ be a non-hyperbolic zero of $V$. Can we perturb $V$ slightly so ...
11
votes
1answer
129 views
+100

Newton iteration on Riemannian manifolds

Suppose $f:M \to N$ is a smooth map between complete Riemannian manifolds of the same dimension. Suppose $Df(m_0)$ is invertible, and $n$ is a point close to $f(m_0)$. Can we perform Newton iteration ...
0
votes
1answer
24 views

Representing a vector field locally

A couple of questions occured to me when reading about the Poincare-Hopf theorem. Any pointers or comments would be appreciated! Let $M$ be a closed oriented Riemannian manifold and $V$ a vector ...